evaluate 6^1/2.6^1/4.6^1/8.....​  

Dear Student,

$$\begin{gathered} 6^{\frac{1}{2}}\times 6^{\frac{1}{4}}\times 6^{\frac{1}{8}}.....=6^{\frac{1}{2}}\times 6^{\frac{1}{2^2}}\times 6^{\frac{1}{2^3}}..... \\ =6^{\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+....} \\ =6^{\frac{{\displaystyle \frac{1}{2}}}{1-{\displaystyle \frac{1}{2}}}} \left(\mathrm{sum} \mathrm{of} \mathrm{infinte} \mathrm{G}.\mathrm{P}.=\frac{\mathrm{a}}{1-\mathrm{r}}\right) \\ =6^{\frac{{\displaystyle \frac{1}{2}}}{{\displaystyle \frac{1}{2}}}} \\ =6 \\ \Rightarrow {\mathbf{6}}^{\frac{\mathbf{1}}{\mathbf{2}}}\mathbf{\times }{\mathbf{6}}^{\frac{\mathbf{1}}{\mathbf{4}}}\mathbf{\times }{\mathbf{6}}^{\frac{\mathbf{1}}{\mathbf{8}}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{=}\mathbf{6} \end{gathered}$$

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